Squares and Square Roots

The square of 98 is $(100 - 2)^2 = 100^2 - 2(100)(2) + 2^2 = 10000 - 400 + 4 = 9604$. Alternatively, $98 \times 98 = 9604$. Other options are incorrect arithmetic results.

93 squared = 93 × 93 = 8649. You can calculate this as (90+3)² = 8100 + 540 + 9 = 8649. The other options are incorrect calculations.

The square of 103 is $(100 + 3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609$. Option 10309 is a common mistake where the middle term of the binomial expansion is omitted.

131 squared = 131 × 131 = 17161. This can be calculated as (130+1)² = 16900 + 260 + 1 = 17161. The other options are incorrect.

The square of 35 is calculated as $35 \times 35$. A shortcut for numbers ending in 5 is to multiply the first digit (3) by the next integer (4) to get 12, and append 25, resulting in 1225. Other options are mathematically incorrect.

In a 9x9 grid, total rectangles = (9×10/2)² = (45)² = 2025. Total squares = 9² + 8² + ... + 1² = 285. Rectangles that are NOT squares = 2025 - 285 = 1740. The claimed answer E (1740) is correct after verification.

In this 4x4 grid, the sum of each row is 44: (15+3+7+19=44), (8+11+5+20=44? No). Looking at columns: 15+8+5+14=42, 3+11+2+7=23. The logic is the sum of the first three numbers in a row equals the fourth? 15+3+7=25 (no). 14+7+14=35. If the sum is 56? 35+21=56. 21 is the intended answer.

A standard chess board is an 8x8 grid, resulting in 64 individual squares. While there are more squares if you count larger composite squares (like 2x2 or 3x3), the question refers to the basic grid units.

Pythagoras and his followers considered 10 to be a 'perfect number' because it represented the sum of the first four numbers (1+2+3+4=10) and could be represented as a triangular figure (tetractys). The number 10 had mystical significance in Pythagorean philosophy. Modern mathematics defines perfect numbers differently (like 6, 28, 496), but in Pythagorean context, 10 is correct.

A standard chessboard consists of an 8x8 grid, totaling 64 squares of alternating colors. This arrangement has been consistent for centuries and is fundamental to the game's strategy. The other numbers (36, 48, 72) do not match the 8x8 configuration.

The LCM of 21 (3x7), 36 (2^2x3^2), and 66 (2x3x11) is 2^2 * 3^2 * 7 * 11 = 2772. To make it a perfect square, we must multiply by 7 and 11 to pair the factors: 2772 * 77 = 213444.

This is a perimeter counting problem. With 27 nails on each side: 4 × 27 = 108. However, each corner nail is counted twice (once for each adjacent side), so we subtract the 4 corner nails: 108 - 4 = 104 nails total.

While there are 64 individual unit squares, the total number of squares of all sizes (1x1, 2x2... 8x8) is calculated as the sum of squares from 1 to 8: 1+4+9+16+25+36+49+64 = 204.

The square of 99 is 9801. You can calculate this as (100 - 1)² = 10000 - 200 + 1 = 9801, or memorize it as 99² = 9801.